PSI - Issue 73

Lenka Koubova et al. / Procedia Structural Integrity 73 (2025) 66–72 Lenka Koubova / Structural Integrity Procedia 00 (2025) 000–000

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5

We can determine the first natural frequency using Eq. (8). = � = � 1ℎ23 ∙ [ ∙ℎ+ ( −1 ) ∙ ] ∙ ∙ ∙ = 1 � 12 ∙ ℎ 3 [ ∙ℎ+ ( −1 ) ∙ ] ∙ ∙

(8)

Fig. 3. Frame structure and its model with one degree of freedom (DoF).

It is clear from Eq. 8 that if we replace the MRF structure with a model with one degree of freedom, the value of the first natural frequency indeed decreases with the inverse value of the number of floors, n f . We can also see that the value of n c , which is the number of columns per floor, can be almost eliminated in the equation.

Fig. 4. The first natural frequency for MRF structures with 5 columns per floor.

The natural frequency is, of course, also dependent on the total weight of the structure. We can increase the total weight of the structure by changing the material to a heavier one, for example, from concrete to steel. Steel has a different modulus of elasticity, E = 210 GPa, and a density, ρ = 7850 kg/m³. We make the change in the parameter in Eq. (8), where the first natural frequency for the single DoF model was derived. , = 1 � 12 ∙ 2 3 1 3 0 ∙ ∙ ℎ 3 [ ∙ℎ+ ( −1 ) ∙ ] ∙ ∙ ∙ 7 2 8 4 9 0 0 0 =1.39 ∙ , (9) The total weight of the structure can be further increased by, for example, increasing the dimension of the square cross-section from 0.4 m to 0.6 m, i.e., the side dimension has increased 1.5 times. , 0 . 6 = 1 � 12 ∙ ∙1 . 5 4 ∙ ℎ 3 [ ∙ℎ+ ( −1 ) ∙ ] ∙ ∙1 . 5 2 ∙ =1.5 ∙ , 0 . 4 (10)